Subgroups — C₁₃×#33692 #15720031

Order 3328.

536 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2238	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2231	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2213	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2190	C₂	1	2	yes	maximal, complemented
7	C₁₃×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₁₃×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#575	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	—	2	1	no	—
11	C₁₃×#540	—	2	1	no	—
12	C₁₃×#509	C₂²	1	4	yes	—
13	C₁₃×#499	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#490	—	2	1	no	—
16	C₁₃×#469	C₂²	1	2	yes	—
17	C₁₃×#466	—	2	1	no	—
18	C₁₃×#444	—	2	1	no	—
19	C₁₃×#444	C₂²	1	2	yes	—
20	C₁₃×#443	—	2	1	no	—
21	C₁₃×#443	C₂²	1	2	yes	—
22	C₁₃×#442	C₂²	1	2	yes	complemented
23	C₄×C₁₃×D₈	—	2	1	no	—
24	C₁₃×#430	C₂²	1	1	yes	characteristic
25	C₁₃×#421	—	2	1	no	—
26	C₁₃×#405	C₂²	1	1	yes	characteristic
27	C₁₃×#365	C₂²	1	4	yes	complemented
28	C₁₃×#364	C₂²	1	2	yes	complemented
29	C₁₃×#361	C₂²	1	4	yes	complemented
30	C₁₃×#360	C₂²	1	4	yes	—
31	C₁₃×#349	C₂²	1	2	yes	—
32	C₁₃×#347	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	2	1	no	—
34	C₁₃×#137	—	2	2	no	—
35	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
36	C₁₃×#135	—	2	1	no	—
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₁₃×#131	—	2	2	no	—
41	C₁₃×#131	—	2	2	no	—
42	C₁₃×#131	—	4	1	no	—
43	C₁₃×#131	—	4	1	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	4	1	no	—
46	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	—	4	1	no	—
48	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#125	—	2	1	no	—
51	C₁₃×#123	—	4	2	no	—
52	C₁₃×#121	—	2	2	no	—
53	C₁₃×#121	—	4	2	no	—
54	C₄×D₄×C₁₃	—	2	2	no	—
55	C₁₃×#117	D₄	1	1	yes	characteristic
56	C₁₃×SD₃₂	—	2	4	no	—
57	C₁₃×D₁₆	—	2	4	no	—
58	C₁₃×M₃₂	C₂³	1	4	yes	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	C₂³	1	2	yes	—
61	C₁₃×#107	—	2	1	no	—
62	C₁₃×#106	C₂³	1	2	yes	—
63	C₁₃×#104	—	2	2	no	—
64	C₁₃×#104	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₁₃×#97	D₄	1	1	yes	characteristic
68	C₁₃×#97	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33692	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	2	2	no	—
73	C₁₃×#41	—	2	2	no	—
74	C₁₃×#41	—	4	1	no	—
75	C₁₃×#41	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	—	2	1	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	#41	1	2	yes	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	—	2	4	no	—
91	C₁₃×D₈	—	4	1	no	—
92	C₁₃×D₈	—	4	1	no	—
93	C₁₃×D₈	#41	1	2	yes	—
94	C₁₃×M₁₆	—	2	2	no	—
95	C₁₃×M₁₆	—	2	2	no	—
96	C₁₃×M₁₆	—	4	2	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	2	no	—
99	C₂×C₈×C₁₃	—	2	1	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	4	2	no	—
102	C₂×C₈×C₁₃	—	4	1	no	—
103	C₂×C₈×C₁₃	—	4	2	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₁₃×C₁₆	—	2	4	no	—
117	#3176	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2955	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2238	C₂×C₁₃	1	2	yes	complemented
120	#2231	C₂×C₁₃	1	2	yes	complemented
121	#2213	C₂×C₁₃	1	2	yes	complemented
122	#2190	C₂×C₁₃	1	2	yes	complemented
123	AGL(1,ℤ/16)	C₂×C₁₃	1	4	yes	complemented
124	#2166	C₂×C₁₃	1	1	yes	characteristic, complemented
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	4	1	no	—
130	D₄×C₁₃	—	2	2	no	—
131	D₄×C₁₃	—	4	1	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	16	2	no	—
137	D₄×C₁₃	—	16	2	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	16	2	no	—
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	4	2	no	—
152	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₈×C₁₃	C₂×#41	1	2	yes	—
155	#575	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#131	—	2	1	no	—
157	#540	—	2	1	no	—
158	#509	C₂²×C₁₃	1	4	yes	—
159	#499	C₂²×C₁₃	1	1	yes	characteristic, complemented
160	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
161	#490	—	2	1	no	—
162	#469	C₂²×C₁₃	1	2	yes	—
163	#466	—	2	1	no	—
164	#444	—	2	1	no	—
165	#444	C₂²×C₁₃	1	2	yes	—
166	#443	—	2	1	no	—
167	#443	C₂²×C₁₃	1	2	yes	—
168	#442	C₂²×C₁₃	1	2	yes	complemented
169	C₄×D₈	—	2	1	no	—
170	#430	C₂²×C₁₃	1	1	yes	characteristic
171	#421	—	2	1	no	—
172	#405	C₂²×C₁₃	1	1	yes	characteristic
173	#365	C₂²×C₁₃	1	4	yes	complemented
174	#364	C₂²×C₁₃	1	2	yes	complemented
175	#361	C₂²×C₁₃	1	4	yes	complemented
176	#360	C₂²×C₁₃	1	4	yes	—
177	#349	C₂²×C₁₃	1	2	yes	—
178	#347	C₂²×C₁₃	1	2	yes	complemented
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	1	no	—
189	C₄×C₁₃	—	4	1	no	—
190	C₄×C₁₃	—	4	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	#525	1	1	yes	characteristic
193	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂×#41	—	2	1	no	—
195	#137	—	2	2	no	—
196	AGL(1,ℤ/8)	—	2	2	no	—
197	#135	—	2	1	no	—
198	#135	C₂³×C₁₃	1	2	yes	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	C₂³×C₁₃	1	2	yes	—
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	1	no	—
204	#131	—	4	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	1	no	—
207	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	—	4	1	no	—
209	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
210	#127	—	2	1	no	—
211	#125	—	2	1	no	—
212	#123	—	4	2	no	—
213	#121	—	2	2	no	—
214	#121	—	4	2	no	—
215	C₄×D₄	—	2	2	no	—
216	#117	D₄×C₁₃	1	1	yes	characteristic
217	SD₃₂	—	2	4	no	—
218	D₁₆	—	2	4	no	—
219	M₃₂	C₂³×C₁₃	1	4	yes	—
220	#108	—	2	1	no	—
221	#108	C₂³×C₁₃	1	2	yes	—
222	#107	—	2	1	no	—
223	#106	C₂³×C₁₃	1	2	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#97	D₄×C₁₃	1	1	yes	characteristic
229	#97	C₂³×C₁₃	1	1	yes	characteristic
230	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	4	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#3109	1	1	yes	characteristic
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	Q₁₆	—	2	1	no	—
251	Q₁₆	C₁₃×#41	1	2	yes	—
252	SD₁₆	—	2	2	no	—
253	SD₁₆	—	2	2	no	—
254	D₈	—	2	1	no	—
255	D₈	—	2	4	no	—
256	D₈	—	4	1	no	—
257	D₈	—	4	1	no	—
258	D₈	C₁₃×#41	1	2	yes	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₁₆	—	2	4	no	—
282	C₁₃	#33692	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	—	4	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	16	2	no	—
295	D₄	—	16	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₈	C₂×C₁₃×#41	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₁₃×#525	1	1	yes	characteristic
327	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	8	2	no	—
331	C₂	—	16	2	no	—
332	C₂	C₁₃×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₁₃×#3176	C₂	semidirect
2. C₁₃×#2955	C₂	semidirect
3. C₁₃×#2238	C₂	semidirect
4. C₁₃×#2231	C₂	semidirect
5. C₁₃×#2213	C₂	semidirect
6. C₁₃×#2190	C₂	semidirect
7. C₁₃×AGL(1,ℤ/16)	C₂	semidirect
8. C₁₃×#2166	C₂	semidirect
13. C₁₃×#499	C₂²	semidirect
14. C₁₃×#493	C₂²	semidirect
22. C₁₃×#442	C₂²	semidirect
27. C₁₃×#365	C₂²	semidirect
28. C₁₃×#364	C₂²	semidirect
29. C₁₃×#361	C₂²	semidirect
32. C₁₃×#347	C₂²	semidirect
71. #33692	C₁₃	direct (N × H)
117. #3176	C₂×C₁₃	semidirect
118. #2955	C₂×C₁₃	semidirect
119. #2238	C₂×C₁₃	semidirect
120. #2231	C₂×C₁₃	semidirect
121. #2213	C₂×C₁₃	semidirect
122. #2190	C₂×C₁₃	semidirect
123. AGL(1,ℤ/16)	C₂×C₁₃	semidirect
124. #2166	C₂×C₁₃	semidirect
159. #499	C₂²×C₁₃	semidirect
160. #493	C₂²×C₁₃	semidirect
168. #442	C₂²×C₁₃	semidirect
173. #365	C₂²×C₁₃	semidirect
174. #364	C₂²×C₁₃	semidirect
175. #361	C₂²×C₁₃	semidirect
178. #347	C₂²×C₁₃	semidirect
282. C₁₃	#33692	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#3176	2
C₁₃×#2955	2
C₁₃×#2238	2
C₁₃×#2231	2
C₁₃×#2213	2
C₁₃×#2190	2
C₁₃×AGL(1,ℤ/16)	2
C₁₃×#2166	2
#33692	13

Subgroups — C13×#33692 #15720031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33692 #15720031